Normed Division Algebra - Classification

Classification

The classification of real division algebras began with Georg Frobenius, continued with Hurwitz and was set in general form by Max Zorn. A brief historical summary may be found in Badger.

A full proof can be found in Kantor and Solodovnikov, and in Shapiro. As a basic idea, if an algebra A is proportional to 1 then it is isomorphic to the real numbers. Otherwise we extend the subalgebra isomorphic to 1 using the Cayley–Dickson construction and introducing a vector e which is orthogonal to 1. This subalgebra is isomorphic to the complex numbers. If this is not all of A then we once again use the Cayley–Dickson construction and another vector orthogonal to the complex numbers and get a subalgebra isomorphic to the quaternions. If this is not all of A then we double up once again and get a subalgebra isomorphic to the Cayley numbers (or Octonions). We now have a theorem which says that every subalgebra of A that contains 1 and is not A is associative. The Cayley numbers are not associative and therefore must be A.